Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Linearized Einstein Equations with Sources on Compact Riemannian Manifolds With Boundary

Roee Leder

TL;DR

The paper addresses the solvability and uniqueness of the linearized Einstein equations with sources on compact Riemannian manifolds with boundary under a Ricci-boundary constraint. It introduces lifted divergence $\textdelta_g$ and a lifted Einstein operator $\mathpzc{D}_{\Gamma}\mathpzc{Ein}_g$, assembling them within a generalized Hodge theory framework on manifolds with boundary via the Boutet de Monvel calculus. The main result characterizes solvability by the conditions $\textdelta_g T = 0$ and $T|_{\partial M}=0$, with the solution unique modulo finite-dimensional obstructions captured by a lifted cohomology; in the typical boundary setting these obstructions vanish, yielding convergence to solvability. The approach blends Green's formulae, Hodge decompositions, and a unique continuation lemma (Aronszajn–Cordes) to remove interior/topological restrictions and to enable a linear-to-nonlinear perturbation strategy for Einstein equations with rough interior geometry. The methods extend Lorentzian results to a Riemannian interior, offering a robust linear framework for handling Einstein equations with boundary data and arbitrary interior geometry.

Abstract

We consider compact Riemannian manifolds with arbitrary interior geometry, assuming only that the Ricci tensor vanishes on the boundary. In this setting, we prove that, under the linear Bianchi gauge and extended Cauchy data, the linearized Einstein equations with sources are globally solvable if and only if the source lies in the kernel of a lifted divergence operator and vanishes on the boundary; the solution is unique. This is the first result of its kind without rigid structural restrictions on topology, geometry, or source. The proof combines generalized Hodge theory with unique continuation results.

The Linearized Einstein Equations with Sources on Compact Riemannian Manifolds With Boundary

TL;DR

The paper addresses the solvability and uniqueness of the linearized Einstein equations with sources on compact Riemannian manifolds with boundary under a Ricci-boundary constraint. It introduces lifted divergence and a lifted Einstein operator , assembling them within a generalized Hodge theory framework on manifolds with boundary via the Boutet de Monvel calculus. The main result characterizes solvability by the conditions and , with the solution unique modulo finite-dimensional obstructions captured by a lifted cohomology; in the typical boundary setting these obstructions vanish, yielding convergence to solvability. The approach blends Green's formulae, Hodge decompositions, and a unique continuation lemma (Aronszajn–Cordes) to remove interior/topological restrictions and to enable a linear-to-nonlinear perturbation strategy for Einstein equations with rough interior geometry. The methods extend Lorentzian results to a Riemannian interior, offering a robust linear framework for handling Einstein equations with boundary data and arbitrary interior geometry.

Abstract

We consider compact Riemannian manifolds with arbitrary interior geometry, assuming only that the Ricci tensor vanishes on the boundary. In this setting, we prove that, under the linear Bianchi gauge and extended Cauchy data, the linearized Einstein equations with sources are globally solvable if and only if the source lies in the kernel of a lifted divergence operator and vanishes on the boundary; the solution is unique. This is the first result of its kind without rigid structural restrictions on topology, geometry, or source. The proof combines generalized Hodge theory with unique continuation results.
Paper Structure (23 sections, 29 theorems, 184 equations)

This paper contains 23 sections, 29 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Generalized Hodge theory
  5. Unique continuation
  6. Structure of the paper
  7. Generalized Hodge theory in a nutshell
  8. Boutet de Monvel calculus
  9. Overdetermined ellipticity
  10. Generalized Hodge theory
  11. Hodge theory for the Einstein equations
  12. Setup
  13. Green's formulae
  14. Einstein constraint equations
  15. Einstein equations
  16. ...and 8 more sections

Key Result

Theorem 1

Let $(M,g)$ be a compact Riemannian manifold with nonempty boundary satisfying $\mathrm{Ric}_{g}|_{{\partial M}}=0$. Given $T\in S^{2}_{M}$, the problem admits a unique solution $\sigma \in S^2_M$ if and only if

Theorems & Definitions (47)

  • Theorem 1
  • Lemma 2
  • Definition 3: Trace operator
  • Definition 4: Normal system of trace operators
  • Proposition 5
  • Definition 6: Elliptic pre-complex, Dirichlet conditions
  • Theorem 7: Lifted complex
  • Theorem 8: Hodge decomposition
  • Theorem 9: Cohomological formulation
  • Definition 10: Disrupted elliptic pre-complex
  • ...and 37 more